Of Trigonometrical Quantities from one another. 155 



(2.) The log. cosine corresponding to log. sin 9'99835 15861 

 is required ? 



Given sine, 



Next less in Table, 



tan^r 



S6C.T 



sin (x + 



A log sin x 



logq 

 log q + tf 

 log q + z" 



=9-9983515861 

 = 9'9983442260 



- 0-0000073601 



logsin(.r 

 log sin x (85) 



A log sin x 



Log. 



11-0580482 



11-0597040 



9-9983516 



6-8668887 



= 4-9829875 sf = -0009616... First approx. 

 4-9839491 *' = -0009637... Second. 

 4-9839512 ^" - -0009637210 = A log cos x 

 8-9402960083 = log cos x 



The cosine required 8'9393322873 = log cos (x + 



The cosine required is that of 85 0' 40" ; its correct value is 

 8'9393322838, which differs from its value computed by our for- 

 mulae in the last two of the ten figures. A more correct result 

 would have been obtained, if the given sine and the next less in 

 the table had been continued to more decimal places, a small 

 change in the sine of so large an angle as 85, producing a con- 

 siderable change in the cosine. This is an inconvenience in cal- 

 culation which can only be obviated by more extensive tables. 



(3.) Let it now be required to determine, by BRIGGS'S 

 Table ( Trigonometria Britannica), the cosine corresponding to 

 sin (ac + A a?) = 9*99993516626353, which is known to be the sine 

 of 89 0' 36'. 



u 2 



